134 BELL SYSTEM TECHNICAL JOURNAL 



as before, and G{ro , r) now denotes the Green's function (5-5). We define 

 Juhy 



Jh= f g{r)P{r)[F(r) - le'''" sin d\dS 



(A4-7) 

 -kK2-k)-' II g(r)P(r)g(r,)P(ro)G(r, , r) dS, dS. 



J H is stationary with respect to small variations in P{r) when P{r) 

 satisfies the integral equation (A4-6). Furthermore, from the integral 

 (5-7) for Rh , 



Rh = iK-(Tc)~'^ [Stationary value of Jh] (A4-8) 



which may be used in the same way as equation (A4-5) for Re ■ 



J. Schwinger has used variational methods with considerable success to 

 deal with obstacles in wave guides.* However, his variational equations 

 differ somewhat from those given here. Some light on the relation between 

 Schwinger's equations and the present one may be obtained by returning 

 to the simple algebraic equations (A4-1) and (A4-2). A rough analogue 

 of the expression required to be stationary in Schwinger's theory is 



(aii.vi + 2ai2.vi-Vo -|- a22.V2)/(^i-Vi + ^2X2)' (A4-9) 



The essential point here is that the stationary value of the expression 

 corresponding to (A4-9) gives the value of an impedance or combination 

 of impedances appearing in some equivalent circuit. Expression (A4-9) 

 may be obtained by expressing /, defined by (A4-2), as a function of .vi 

 and y = X2/.V1 . / is still to be made stationary but now it is a function of 

 Xi and y. Solving dj/dxi = for Xi and setting this value of .Vi in / gives 

 the following function of y 



-(bi + b^yT- (an + 2aviy + ^22/)-', 



which is the stationary value of / with respect to variations in xi when y is 

 held constant. This function is still required to be stationary with respect 

 to y. The same is true of its reciprocal which becomes (A4-9) when both 

 numerator and denominator are multiplied by .vi and the definition of y 

 used. When (A4-1) is replaced by a larger number of equations similar 

 considerations lead to a generalized form of (A4-9). The expression required 

 to be stationary by Schwinger is obtained when the sums in the general- 

 ized form are replaced by integrals. 



* An account of the method together with applications is given in "Notes on Ix-ctures 

 by Julian Schwinger: Discontinuities in Waveguides" by David S. Saxon. An account 

 is also given by John VV. Miles." 



